Measurable function, $f$, with $\int_{(0,1)} |f(x)| dm(x) < \infty$ and $\int_{(0,1)} (f(x))^{2} dm(x) = \infty$?

Question

I need to think of an example of a measurable function that satisfies \begin{equation} \int_{(0,1)} |f(x)| dm(x) < \infty \quad \text{and} \int_{(0,1)} (f(x))^{2} dm(x) = \infty. \end{equation} I've been thinking about this for a while now and I can't seem to think of an example. It seems rather counterintuitive.

Answer 1

$f(x)=x^{-1/2}$ is one example.